Computable metrics above the standard real metric

We construct a sequence of computable real metrics pairwise incomparable under weak reducibility $\leq_{ch}$ and located above the standard real metric w. r. t. computable reducibility $\leq_c$. Iterating the construction, we obtain that the ordering $(P(\omega),\subseteq)$ of subsets of $\omega$ is embeddable into the ordering of $ch$-degrees of real metrics above the standard metric. It is also proved that the countable atomless Boolean algebra is embeddable with preservation of joins and meets into the ordering of $c$-degrees of computable real metrics.